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Properties

Label 16.0.30611001600...0000.1
Degree $16$
Signature $[0, 8]$
Discriminant $2^{16}\cdot 3^{14}\cdot 5^{10}$
Root discriminant $14.30$
Ramified primes $2, 3, 5$
Class number $2$
Class group $[2]$
Galois group $(C_4\times C_8):C_2$ (as 16T114)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![109, -438, 683, -390, -164, 138, 578, -1110, 862, -210, -199, 216, -83, 0, 14, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 6*x^15 + 14*x^14 - 83*x^12 + 216*x^11 - 199*x^10 - 210*x^9 + 862*x^8 - 1110*x^7 + 578*x^6 + 138*x^5 - 164*x^4 - 390*x^3 + 683*x^2 - 438*x + 109)
 
gp: K = bnfinit(x^16 - 6*x^15 + 14*x^14 - 83*x^12 + 216*x^11 - 199*x^10 - 210*x^9 + 862*x^8 - 1110*x^7 + 578*x^6 + 138*x^5 - 164*x^4 - 390*x^3 + 683*x^2 - 438*x + 109, 1)
 

Normalized defining polynomial

\( x^{16} - 6 x^{15} + 14 x^{14} - 83 x^{12} + 216 x^{11} - 199 x^{10} - 210 x^{9} + 862 x^{8} - 1110 x^{7} + 578 x^{6} + 138 x^{5} - 164 x^{4} - 390 x^{3} + 683 x^{2} - 438 x + 109 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 8]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(3061100160000000000=2^{16}\cdot 3^{14}\cdot 5^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $14.30$
magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
 
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
Ramified primes:  $2, 3, 5$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is not a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{13} a^{14} - \frac{5}{13} a^{13} + \frac{3}{13} a^{12} - \frac{6}{13} a^{11} - \frac{3}{13} a^{10} + \frac{2}{13} a^{9} + \frac{3}{13} a^{8} + \frac{2}{13} a^{7} + \frac{1}{13} a^{6} - \frac{3}{13} a^{5} - \frac{3}{13} a^{4} - \frac{3}{13} a^{3} - \frac{6}{13} a^{2} - \frac{1}{13} a + \frac{3}{13}$, $\frac{1}{51400617151} a^{15} - \frac{574886150}{51400617151} a^{14} - \frac{1283058164}{3953893627} a^{13} - \frac{8671934673}{51400617151} a^{12} + \frac{20293158180}{51400617151} a^{11} + \frac{6090980033}{51400617151} a^{10} - \frac{643500235}{51400617151} a^{9} + \frac{21309473649}{51400617151} a^{8} + \frac{15939295551}{51400617151} a^{7} + \frac{15516788035}{51400617151} a^{6} + \frac{17006656852}{51400617151} a^{5} + \frac{25335860592}{51400617151} a^{4} - \frac{293023523}{3953893627} a^{3} + \frac{8321332667}{51400617151} a^{2} + \frac{14854654885}{51400617151} a + \frac{21573882714}{51400617151}$

magma: IntegralBasis(K);
 
sage: K.integral_basis()
 
gp: K.zk
 

Class group and class number

$C_{2}$, which has order $2$

magma: ClassGroup(K);
 
sage: K.class_group().invariants()
 
gp: K.clgp
 

Unit group

magma: UK, f := UnitGroup(K);
 
sage: UK = K.unit_group()
 
Rank:  $7$
magma: UnitRank(K);
 
sage: UK.rank()
 
gp: K.fu
 
Torsion generator:  \( -\frac{5495145360}{51400617151} a^{15} + \frac{25311322742}{51400617151} a^{14} - \frac{39636540612}{51400617151} a^{13} - \frac{64317809259}{51400617151} a^{12} + \frac{379770233073}{51400617151} a^{11} - \frac{630981304914}{51400617151} a^{10} + \frac{75800548949}{51400617151} a^{9} + \frac{113128199498}{3953893627} a^{8} - \frac{2670681844193}{51400617151} a^{7} + \frac{1817158917442}{51400617151} a^{6} + \frac{250377543978}{51400617151} a^{5} - \frac{892011581055}{51400617151} a^{4} - \frac{549176230526}{51400617151} a^{3} + \frac{1641258595500}{51400617151} a^{2} - \frac{1150135214525}{51400617151} a + \frac{267482962987}{51400617151} \) (order $12$)
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
Fundamental units:  Units are too long to display, but can be downloaded with other data for this field from 'Stored data to gp' link to the right
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 1367.18712937 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$(C_4\times C_8):C_2$ (as 16T114):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 64
The 28 conjugacy class representatives for $(C_4\times C_8):C_2$
Character table for $(C_4\times C_8):C_2$ is not computed

Intermediate fields

\(\Q(\sqrt{-1}) \), \(\Q(\sqrt{3}) \), \(\Q(\sqrt{-3}) \), \(\Q(\zeta_{12})\), 8.0.4665600.1

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

Degree 16 sibling: data not computed
Degree 32 siblings: data not computed

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type R R R ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/41.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.

magma: p := 7; // to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
magma: idealfactors := Factorization(p*Integers(K)); // get the data
 
magma: [<primefactor[2], Valuation(Norm(primefactor[1]), p)> : primefactor in idealfactors];
 
sage: p = 7; # to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
sage: [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]
 
gp: p = 7; \\ to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
gp: idealfactors = idealprimedec(K, p); \\ get the data
 
gp: vector(length(idealfactors), j, [idealfactors[j][3], idealfactors[j][4]])
 

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
2Data not computed
3Data not computed
$5$5.8.6.3$x^{8} + 25 x^{4} + 200$$4$$2$$6$$C_8$$[\ ]_{4}^{2}$
5.8.4.2$x^{8} + 25 x^{4} - 250 x^{2} + 1250$$2$$4$$4$$C_8$$[\ ]_{2}^{4}$